TTP Limit of Detection

Suzanne Dufault

General

Overview (1/2)

  • BACTEC MGIT machine is the standard approach globally for detecting viable bacilli in a patient’s sputum.
    • Measures the decline of oxygen in a liquid culture tube inoculated with the sputum sample.
    • Introduced during a major diagnostic development push at the end of the 1980s/early 1990s.
  • Samples observed no longer than 42 days, at which point the sample is declared ``negative’’ for TB with no evidence for bacterial growth.

Overview (2/2)

In practice, very few samples return MGIT time-to-positivity values between 30 and 42 days. While this range may be important for individual diagnostic purposes, the TTP values captured in this range may not contribute meaningfully to understanding the relative TTP trajectories at the regimen-level. We investigate whether using a lower limit of detection for longitudinal time-to-positivity modeling will improve precision for detecting differences between treatments, resulting in shorter trial times and improved power for performing regimen selection.

Objective

Our objective is to examine existing TTP data from various case studies to better define the trajectory and variance of regimen-level TTP over time, while examining the impact of alternate limits of detection on power.

Datasets

  • REMoxTB

  • PanACEA MAMS-TB

REMoxTB

We will restrict to only those with:

  • result = 0,1,2 (i.e., only those with a Negative (0), Positive (1), or Unconfirmed Positive (2) result)
  • bact = 1 (i.e., MGIT)
  • weeks <= 8 (i.e., only looking at observations within 8 weeks of randomization)
  • !box_mitt %in% c(10,20,30) excluding those with late screening failures

MAMS

We restrict to those with:

  • WEEK <= 8 (i.e., only looking at observations within 8 weeks of randomization)

  • !is.na(DV) (i.e., only those with an observed time to positivity)

TB-PACTS NC-002 M-Pa-Z

<environment: R_GlobalEnv>

TB-PACTS NC-005 M-Pa-Z-B

TB-PACTS NC-006 (STAND)

TB-PACTS TBTC-S29

TB-PACTS TBTC-S29x

Methods - Models

  • Models: Bayesian (linear fit), Burger bi-linear model, etc.

  • LOQ: 25, 30, 42

Descriptive Profile

Quick Summary

REMoxTB
ttp n prop
[0,30) 11058 0.7505
[30,42) 221 0.0150
[42,45) 3455 0.2345
MAMS-TB
ttp n prop
[0,30) 2265 0.7325
[30,42) 117 0.0378
[42,45) 710 0.2296
REMoxTB
week ttp n prop
0 [0,30) 3291 0.9213
0 [30,42) 5 0.0014
0 [42,45) 276 0.0773
1 [0,30) 1434 0.8792
1 [30,42) 5 0.0031
1 [42,45) 192 0.1177
2 [0,30) 1327 0.8656
2 [30,42) 13 0.0085
2 [42,45) 193 0.1259
3 [0,30) 1237 0.8313
3 [30,42) 13 0.0087
3 [42,45) 238 0.1599
4 [0,30) 1086 0.7521
4 [30,42) 29 0.0201
4 [42,45) 329 0.2278
5 [0,30) 939 0.6726
5 [30,42) 29 0.0208
5 [42,45) 428 0.3066
6 [0,30) 792 0.5781
6 [30,42) 47 0.0343
6 [42,45) 531 0.3876
7 [0,30) 621 0.4510
7 [30,42) 52 0.0378
7 [42,45) 704 0.5113
8 [0,30) 331 0.3586
8 [30,42) 28 0.0303
8 [42,45) 564 0.6111
MAMS-TB
week ttp n prop
0 [0,30) 394 0.9975
0 [30,42) 1 0.0025
1 [0,30) 387 0.9724
1 [30,42) 2 0.0050
1 [42,45) 9 0.0226
2 [0,30) 344 0.9198
2 [30,42) 4 0.0107
2 [42,45) 26 0.0695
3 [0,30) 314 0.9075
3 [30,42) 6 0.0173
3 [42,45) 26 0.0751
4 [0,30) 252 0.7522
4 [30,42) 17 0.0507
4 [42,45) 66 0.1970
5 [0,30) 209 0.6276
5 [30,42) 29 0.0871
5 [42,45) 95 0.2853
6 [0,30) 164 0.5325
6 [30,42) 17 0.0552
6 [42,45) 127 0.4123
7 [0,30) 112 0.3836
7 [30,42) 22 0.0753
7 [42,45) 158 0.5411
8 [0,30) 89 0.2862
8 [30,42) 19 0.0611
8 [42,45) 203 0.6527

TTP v. Time (REMoxTB)

Observed TTP trajectories, where anything at or above 42 receives a value of 42.

TTP v. Time (REMoxTB) Takeaway

Panel A

  • The two novel regimen trajectories are indistinguishable from each other.
  • Both novel regimens appear to improve TTP faster than control.

Panel B

  • Individual trajectories are often high variance
  • Individual trajectories tend to be increasing over time. There are very few that appear to start high/low and stay there. (e.g., treatment effect is observable)

TTP v. Time (MAMS-TB)

Observed TTP trajectories, where anything at or above 42 receives a value of 42.

TTP v. Time (MAMS-TB) Takeaway

Panel A

  • There is substantial overlap between all regimens, but all appear to have steeper slope (at least initially) during the first 4 weeks.

Panel B

  • Individual trajectories are often high variance
  • Individual trajectories tend to be increasing over time. There are very few that appear to start high/low and stay there. (e.g., treatment effect is observable)

Data Distribution v. Time

[WANT ALLUVIAL PLOT… ]

Weekly Alluvial Plot - REMoxTB

Weekly Alluvial Plot - MAMS-TB

Alluvial Plots: Takeaways

  • “Most” people convert from TTP less than 30 to above the limit of detection without a sample returning a TTP between 30 and 42 days.
  • The 30-42 day observations tend to occur in weeks 4+
  • NOTE by week 8, the majority of samples are not returning any information.

Weekly TTP Distribution (REMoxTB)

Distribution of TTP measures at each week post-randomization.

Takeaway: Weekly TTP Distribution (REMoxTB)

Variance/Signal-to-Noise Investigation

Proportion of data censored for varying LOQ limits

Idea: Maybe this should a “relative” to 42 plot, rather than an absolute plot? Would that make it easier to read? or add in limited horizontal lines?

Table Censored (REMoxTB)

Proportion of samples below censoring limit
week 25 days 30 days 35 days 42 days Difference in proportions: 42 v 30 days
0 0.9197 0.9213 0.9225 0.9227 0.0014
1 0.8743 0.8792 0.8792 0.8823 0.0031
2 0.8565 0.8656 0.8689 0.8741 0.0085
3 0.8132 0.8313 0.8367 0.8401 0.0087
4 0.7216 0.7521 0.7632 0.7722 0.0201
5 0.6426 0.6726 0.6862 0.6934 0.0208
6 0.5350 0.5781 0.5978 0.6124 0.0343
7 0.4110 0.4510 0.4742 0.4887 0.0378
8 0.3109 0.3586 0.3749 0.3889 0.0303

Table Censored (MAMS)

Proportion of samples below censoring limit
week 25 days 30 days 35 days 42 days Difference in proportions: 42 v 30 days
0 0.9975 0.9975 0.9975 1.0000 0.0025
1 0.9623 0.9724 0.9749 0.9774 0.0050
2 0.9011 0.9198 0.9305 0.9305 0.0107
3 0.8815 0.9075 0.9162 0.9249 0.0173
4 0.7045 0.7522 0.7940 0.8030 0.0507
5 0.5856 0.6276 0.6607 0.7147 0.0871
6 0.4610 0.5325 0.5519 0.5877 0.0552
7 0.3288 0.3836 0.4315 0.4589 0.0753
8 0.2444 0.2862 0.3087 0.3473 0.0611

Takeaway

  • Changing the censoring limit to 30 or 35 has very limited impact on the proportion of data “left out” of the analyses, especially at early weeks where the data is the richest.

Modeling Work

MAMS-TB: Linear (treatment level)

(MAMS-TB) Posterior group-level estimates of slope from the 30- and 42-day models (95% Credible Intervals).
Estimate CI.l CI.u CI width Point Estimation CI Estimation
LOQ = 42
HR20ZM.weeks 0.138 0.124 0.154 0.030 mode hdi
HR20ZQ.weeks 0.121 0.104 0.137 0.033 mode hdi
HR35ZE.weeks 0.144 0.126 0.160 0.034 mode hdi
HRZE.weeks 0.128 0.116 0.138 0.022 mode hdi
HRZQ.weeks 0.122 0.104 0.137 0.033 mode hdi
LOQ = 30
HR20ZM.weeks 0.132 0.117 0.146 0.029 mode hdi
HR20ZQ.weeks 0.116 0.101 0.130 0.029 mode hdi
HR35ZE.weeks 0.138 0.123 0.155 0.032 mode hdi
HRZE.weeks 0.122 0.109 0.131 0.022 mode hdi
HRZQ.weeks 0.115 0.098 0.129 0.031 mode hdi
LOQ = 25
HR20ZM.weeks 0.127 0.114 0.142 0.028 mode hdi
HR20ZQ.weeks 0.117 0.099 0.129 0.030 mode hdi
HR35ZE.weeks 0.137 0.121 0.155 0.034 mode hdi
HRZE.weeks 0.120 0.109 0.131 0.022 mode hdi
HRZQ.weeks 0.115 0.097 0.128 0.031 mode hdi

REMoxTB: Linear (treatment level)

(REMoxTB) Posterior group-level estimates of slope from the 30- and 42-day models.
Estimate CI.l CI.u CI width Point Estimation CI Estimation
LOQ = 42
1. 2EHRZ/4HR.weeks 0.095 0.090 0.100 0.010 mode hdi
2. 2MHRZ/2MHR.weeks 0.104 0.100 0.109 0.009 mode hdi
3. 2EMRZ/2MR.weeks 0.107 0.102 0.111 0.009 mode hdi
LOQ = 30
1. 2EHRZ/4HR.weeks 0.089 0.085 0.093 0.008 mode hdi
2. 2MHRZ/2MHR.weeks 0.097 0.094 0.102 0.008 mode hdi
3. 2EMRZ/2MR.weeks 0.099 0.095 0.103 0.008 mode hdi
LOQ = 25
1. 2EHRZ/4HR.weeks 0.086 0.082 0.090 0.008 mode hdi
2. 2MHRZ/2MHR.weeks 0.095 0.091 0.099 0.008 mode hdi
3. 2EMRZ/2MR.weeks 0.097 0.093 0.100 0.007 mode hdi

Forest plot

Linear changes in estimates

(MAMS-TB) Change in point estimate and confidence interval width (30 days / 42 days).
key point.estimate.ratio.30.42 CI.width.ratio.30.42 point.estimate.ratio.25.42 CI.width.ratio.25.42
HR20ZM.weeks 0.9565 0.9667 0.9203 0.9333
HR20ZQ.weeks 0.9587 0.8788 0.9669 0.9091
HR35ZE.weeks 0.9583 0.9412 0.9514 1.0000
HRZE.weeks 0.9531 1.0000 0.9375 1.0000
HRZQ.weeks 0.9426 0.9394 0.9426 0.9394
(REMoxTB) Change in point estimate and confidence interval width (30 days / 42 days).
key point.estimate.ratio CI.width.ratio
1. 2EHRZ/4HR.weeks 0.9368 0.8000
2. 2MHRZ/2MHR.weeks 0.9327 0.8889
3. 2EMRZ/2MR.weeks 0.9252 0.8889

MAMS-TB: Biphasic (treatment level)

(MAMS-TB) Posterior group-level estimates of slope from the 30- and 42-day models.
regimen coefficient Estimate CI.l CI.u CI width Point Estimation CI Estimation LOQ
beta_1
HR20ZM beta1 0.000 -1.559 0.221 1.780 mode qi 42
HR20ZM beta1 -0.825 -0.829 0.272 1.101 mode qi 30
HR20ZQ beta1 0.125 -0.821 0.216 1.037 mode qi 42
HR20ZQ beta1 -0.821 -0.825 0.264 1.089 mode qi 30
HR35ZE beta1 0.689 -0.763 0.689 1.452 mode qi 42
HR35ZE beta1 -0.788 -0.790 0.276 1.066 mode qi 30
HRZE beta1 0.118 0.116 1.253 1.137 mode qi 42
HRZE beta1 -0.797 -0.800 0.275 1.075 mode qi 30
HRZQ beta1 0.121 0.122 1.350 1.228 mode qi 42
HRZQ beta1 -0.809 -0.813 0.265 1.078 mode qi 30
beta_2
HR20ZM beta2 -0.146 -1.707 -0.052 1.655 mode qi 42
HR20ZM beta2 -0.973 -0.976 -0.032 0.944 mode qi 30
HR20ZQ beta2 -0.208 -0.937 -0.053 0.884 mode qi 42
HR20ZQ beta2 -0.933 -0.938 -0.077 0.861 mode qi 30
HR35ZE beta2 0.519 -0.913 0.518 1.431 mode qi 42
HR35ZE beta2 -0.933 -0.936 -0.054 0.882 mode qi 30
HRZE beta2 1.136 -0.478 1.136 1.614 mode qi 42
HRZE beta2 -0.927 -0.928 -0.041 0.887 mode qi 30
HRZQ beta2 -0.212 -0.208 1.240 1.448 mode qi 42
HRZQ beta2 -0.927 -0.929 -0.058 0.871 mode qi 30

REMoxTB: Biphasic (treatment level)

(REMoxTB) Posterior group-level estimates of slope from the 30- and 42-day models.
regimen coefficient Estimate CI.l CI.u CI width Point Estimation CI Estimation LOQ
beta_1
2EHRZ/4HR beta1 0.098 0.090 0.219 0.129 mode qi 42
2EHRZ/4HR beta1 0.092 0.017 0.233 0.216 mode qi 30
2EMRZ/2MR beta1 0.098 0.096 0.225 0.129 mode qi 42
2EMRZ/2MR beta1 0.098 0.010 0.236 0.226 mode qi 30
2MHRZ/2MHR beta1 0.106 0.097 0.219 0.122 mode qi 42
2MHRZ/2MHR beta1 0.093 0.050 0.229 0.179 mode qi 30
beta_2
2EHRZ/4HR beta2 -0.140 -0.370 -0.088 0.282 mode qi 42
2EHRZ/4HR beta2 -0.129 -1.346 -0.084 1.262 mode qi 30
2EMRZ/2MR beta2 -0.128 -0.365 -0.097 0.268 mode qi 42
2EMRZ/2MR beta2 -0.109 -1.281 -0.088 1.193 mode qi 30
2MHRZ/2MHR beta2 -0.126 -0.374 -0.098 0.276 mode qi 42
2MHRZ/2MHR beta2 -0.113 -1.401 -0.096 1.305 mode qi 30

ALL (visualization)

Takeaway

For the REMoxTB and MAMS-TB data, point estimates in the 30-day models decrease relative to the 42-day models, as expected when “removing” data at the upper end of the spectrum.

However, credible intervals (e.g., precision) remain either the same or improve in the 30-day models relative to the 42-day models. In the REMoxTB data, the CIs improve more than the point estimates shrink. This is also true for 3 out of 5 arms in the MAMS-TB data.

Differentiation of Regimens

Motivation

We’re not just interested in the estimation of TTP slopes. More importantly, we want to directly address whether improved estimation allows us to make “better” decisions, ideally earlier. Therefore, this section will look at the ability to differentiate “promising” regimens relative to control, where promising is assumed to mean those with steeper changes in log10(TTP) than control regimens.

MAMS

# A tibble: 3 × 2
  which.LOQ.best `n()`
  <chr>          <int>
1 25 days           24
2 30 days           27
3 42 days            9
# A tibble: 3 × 2
  which.LOQ.best `n()`
  <chr>          <int>
1 25 days            3
2 30 days            3
3 42 days            6

Takeaway

For the MAMS-TB data, using a 30-day limit of quantification improves the ability to differentiate regimens relative to the control. The estimated posterior probability (e.g., “confidence”) that a given regimen’s rate of change is as steep or steeper than then the control regimen’s rate of change is best for nearly all regimens with a 30- rather than 42-day limit of quantification.

REMoxTb

Takeaway

The same trend as was observed in MAMS is not observed in the REMoxTB data, which is perplexing. The 42-day limit of quantification is observed to correlate with a stronger posterior probability of differentiation between the novel regimens and the control (HRZE).

Should we be doing more of a grid-search or can we motivate this better with a S-N exploration first?

Signal-to-Noise

Replicates (NC-002)

Replicates over time (NC-002)

Not super satisfied with this plot, but I do like that it seems to confirm that the deviance in replicates increases as the expected value of TTP increases (e.g., with time).

Sensitivity Analyses

MAMS-TB: Linear

(MAMS-TB) Posterior group-level estimates of slope from the 30- and 42-day models (95% Credible Intervals).
Estimate CI.l CI.u CI width Point Estimation CI Estimation
LOQ = 42
HR20ZM.weeks 0.118 0.105 0.136 0.031 mode hdi
HR20ZQ.weeks 0.109 0.089 0.121 0.032 mode hdi
HR35ZE.weeks 0.124 0.108 0.142 0.034 mode hdi
HRZE.weeks 0.113 0.101 0.126 0.025 mode hdi
HRZQ.weeks 0.110 0.091 0.124 0.033 mode hdi
LOQ = 30
HR20ZM.weeks 0.110 0.096 0.125 0.029 mode hdi
HR20ZQ.weeks 0.101 0.083 0.114 0.031 mode hdi
HR35ZE.weeks 0.117 0.100 0.134 0.034 mode hdi
HRZE.weeks 0.105 0.093 0.116 0.023 mode hdi
HRZQ.weeks 0.102 0.084 0.114 0.030 mode hdi
LOQ = 25
HR20ZM.weeks 0.104 0.091 0.119 0.028 mode hdi
HR20ZQ.weeks 0.097 0.080 0.110 0.030 mode hdi
HR35ZE.weeks 0.111 0.097 0.129 0.032 mode hdi
HRZE.weeks 0.102 0.090 0.113 0.023 mode hdi
HRZQ.weeks 0.098 0.081 0.110 0.029 mode hdi

REMoxTB: Linear

(REMoxTB) Posterior group-level estimates of slope from the 30- and 42-day models (95% Credible Intervals).
Estimate CI.l CI.u CI width Point Estimation CI Estimation
LOQ = 42
1. 2EHRZ/4HR.weeks 0.079 0.073 0.084 0.011 mode hdi
2. 2MHRZ/2MHR.weeks 0.083 0.079 0.088 0.009 mode hdi
3. 2EMRZ/2MR.weeks 0.087 0.081 0.092 0.011 mode hdi
LOQ = 30
1. 2EHRZ/4HR.weeks 0.070 0.065 0.075 0.010 mode hdi
2. 2MHRZ/2MHR.weeks 0.074 0.069 0.077 0.008 mode hdi
3. 2EMRZ/2MR.weeks 0.076 0.072 0.081 0.009 mode hdi
LOQ = 25
1. 2EHRZ/4HR.weeks 0.065 0.061 0.070 0.009 mode hdi
2. 2MHRZ/2MHR.weeks 0.069 0.065 0.073 0.008 mode hdi
3. 2EMRZ/2MR.weeks 0.072 0.067 0.076 0.009 mode hdi

Forest plot

Visualization (ALL)

Model Convergence - MAMS-TB

MAMS-TB: Linear 25-day

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: log10(dtp_25) | cens(censored_25) ~ weeks + (1 + weeks | patient.id + Treatm_arm) 
   Data: df_analysis_mams (Number of observations: 3092) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~patient.id (Number of levels: 363) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.11      0.01     0.09     0.14 1.00     1150
sd(weeks)                0.04      0.00     0.04     0.05 1.01      684
cor(Intercept,weeks)     0.07      0.16    -0.20     0.45 1.01      362
                     Tail_ESS
sd(Intercept)            1993
sd(weeks)                1471
cor(Intercept,weeks)      677

~Treatm_arm (Number of levels: 5) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.02      0.03     0.00     0.08 1.00     1648
sd(weeks)                0.02      0.02     0.00     0.06 1.00      776
cor(Intercept,weeks)    -0.17      0.58    -0.98     0.92 1.00     1488
                     Tail_ESS
sd(Intercept)            2290
sd(weeks)                 896
cor(Intercept,weeks)     2707

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept     0.76      0.02     0.73     0.80 1.00     2955     2264
weeks         0.12      0.01     0.10     0.15 1.00     1100      751

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.26      0.00     0.25     0.27 1.00     2543     2291

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

MAMS-TB: Linear 25-day posterior

MAMS-TB: Linear 30-day

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: log10(dtp_30) | cens(censored_30) ~ weeks + (1 + weeks | patient.id + Treatm_arm) 
   Data: df_analysis_mams (Number of observations: 3092) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~patient.id (Number of levels: 363) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.12      0.01     0.09     0.14 1.01      905
sd(weeks)                0.05      0.00     0.04     0.05 1.02      405
cor(Intercept,weeks)     0.09      0.17    -0.21     0.46 1.03      208
                     Tail_ESS
sd(Intercept)            1463
sd(weeks)                1002
cor(Intercept,weeks)      372

~Treatm_arm (Number of levels: 5) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.02      0.02     0.00     0.08 1.00     1881
sd(weeks)                0.02      0.02     0.00     0.06 1.01      676
cor(Intercept,weeks)    -0.15      0.58    -0.97     0.93 1.00     1024
                     Tail_ESS
sd(Intercept)            2511
sd(weeks)                 748
cor(Intercept,weeks)     1858

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept     0.76      0.02     0.73     0.80 1.00     3045     2321
weeks         0.12      0.01     0.10     0.15 1.00     1433     1001

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.26      0.00     0.26     0.27 1.00     2984     2722

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

MAMS-TB: Linear 30-day posterior

MAMS-TB: Linear 42-day

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: log10(dtp_42) | cens(censored_42) ~ weeks + (1 + weeks | patient.id + Treatm_arm) 
   Data: df_analysis_mams (Number of observations: 3092) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~patient.id (Number of levels: 363) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.13      0.01     0.10     0.15 1.00     1284
sd(weeks)                0.05      0.00     0.04     0.06 1.01      600
cor(Intercept,weeks)    -0.01      0.14    -0.24     0.29 1.01      304
                     Tail_ESS
sd(Intercept)            1779
sd(weeks)                1229
cor(Intercept,weeks)      622

~Treatm_arm (Number of levels: 5) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.02      0.03     0.00     0.09 1.00     1959
sd(weeks)                0.02      0.02     0.00     0.06 1.01      756
cor(Intercept,weeks)    -0.13      0.59    -0.97     0.93 1.00     1352
                     Tail_ESS
sd(Intercept)            2465
sd(weeks)                1230
cor(Intercept,weeks)     2346

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept     0.76      0.02     0.72     0.80 1.00     3390     2943
weeks         0.13      0.01     0.11     0.16 1.00     1528     1009

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.28      0.00     0.27     0.28 1.00     3169     2511

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

MAMS-TB: Linear 42-day posterior

MAMS-TB: Nonlinear 25-day

MAMS-TB: Nonlinear 25-day posterior

MAMS-TB: Nonlinear 30-day

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: log10(dtp_30) | cens(censored_30) ~ alpha + beta1 * weeks + beta2 * gamma * log((exp((weeks - kappa)/gamma) + exp(-(weeks - kappa)/gamma))/(exp(kappa/gamma) + exp(-kappa/gamma))) 
         alpha ~ 1 + (1 | patient.id + Treatm_arm)
         beta1 ~ 1 + (1 | patient.id + Treatm_arm)
         beta2 ~ 1 + (1 | patient.id + Treatm_arm)
         kappa ~ 1 + (1 | patient.id + Treatm_arm)
         gamma ~ 1 + (1 | patient.id + Treatm_arm)
   Data: filter(df_analysis_mams, DV != -99) (Number of observations: 3092) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~patient.id (Number of levels: 363) 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(alpha_Intercept)     0.14      0.01     0.12     0.16 1.80       17       84
sd(beta1_Intercept)     0.03      0.01     0.00     0.04 2.99        5       13
sd(beta2_Intercept)     0.04      0.01     0.03     0.05 1.70        6       33
sd(kappa_Intercept)     0.53      0.42     0.06     1.22 2.18        6       11
sd(gamma_Intercept)     0.14      0.14     0.00     0.48 2.64        6       13

~Treatm_arm (Number of levels: 5) 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(alpha_Intercept)     0.03      0.02     0.00     0.07 1.33       14       39
sd(beta1_Intercept)     0.01      0.01     0.00     0.05 1.21       20       92
sd(beta2_Intercept)     0.02      0.02     0.00     0.10 1.91        6       12
sd(kappa_Intercept)     1.64      0.61     1.08     3.18 1.35       10       79
sd(gamma_Intercept)     0.19      0.22     0.01     0.80 1.59       13       67

Population-Level Effects: 
                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
alpha_Intercept     0.70      0.04     0.63     0.75 2.16        5       23
beta1_Intercept    -0.05      0.43    -0.80     0.27 2.63        5       14
beta2_Intercept    -0.31      0.36    -0.94    -0.04 2.88        5       13
kappa_Intercept     4.31      3.39     2.03    10.17 1.75        6       13
gamma_Intercept     0.57      0.35     0.15     1.36 1.61        7       12

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.25      0.00     0.24     0.26 1.30       11       56

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

MAMS-TB: Nonlinear 30-day posterior

MAMS-TB: Nonlinear 42-day

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: log10(dtp_42) | cens(censored_42) ~ alpha + beta1 * weeks + beta2 * gamma * log((exp((weeks - kappa)/gamma) + exp(-(weeks - kappa)/gamma))/(exp(kappa/gamma) + exp(-kappa/gamma))) 
         alpha ~ 1 + (1 | patient.id + Treatm_arm)
         beta1 ~ 1 + (1 | patient.id + Treatm_arm)
         beta2 ~ 1 + (1 | patient.id + Treatm_arm)
         kappa ~ 1 + (1 | patient.id + Treatm_arm)
         gamma ~ 1 + (1 | patient.id + Treatm_arm)
   Data: df_analysis_mams (Number of observations: 3092) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~patient.id (Number of levels: 363) 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(alpha_Intercept)     0.13      0.01     0.11     0.16 1.90        7       19
sd(beta1_Intercept)     0.04      0.01     0.03     0.05 2.41        5       23
sd(beta2_Intercept)     0.11      0.12     0.04     0.32 2.99        5       11
sd(kappa_Intercept)     0.97      0.81     0.09     2.97 2.74        5       12
sd(gamma_Intercept)     1.92      3.05     0.01     7.28 2.78        5       20

~Treatm_arm (Number of levels: 5) 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(alpha_Intercept)     0.03      0.02     0.00     0.05 1.76        8       40
sd(beta1_Intercept)     0.51      0.63     0.00     2.32 2.04        5       17
sd(beta2_Intercept)     0.58      0.55     0.00     1.82 2.85        5       17
sd(kappa_Intercept)     0.99      0.71     0.14     2.56 1.97        6       13
sd(gamma_Intercept)     5.98      9.15     0.04    21.97 2.79        5       26

Population-Level Effects: 
                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
alpha_Intercept     0.71      0.04     0.66     0.77 2.00        6       21
beta1_Intercept     0.10      0.24    -0.59     0.33 2.23        6       36
beta2_Intercept    -0.05      0.35    -0.71     0.50 3.01        5       13
kappa_Intercept     6.37      3.52     2.07    10.92 2.99        5       37
gamma_Intercept     1.13      0.44     0.29     1.76 2.25        5       31

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.27      0.01     0.26     0.27 1.64        7       40

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

MAMS-TB: Nonlinear 42-day posterior

Model Convergence - REMoxTB

REMoxTB: Linear 25-day

REMoxTB: Linear 25-day posterior

REMoxTB: Linear 30-day

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: log10(dtp_30) | cens(censored_30) ~ weeks + (1 + weeks | trial_no + treat) 
   Data: df_analysis_remox (Number of observations: 14734) 
  Draws: 4 chains, each with iter = 4000; warmup = 2000; thin = 1;
         total post-warmup draws = 8000

Group-Level Effects: 
~treat (Number of levels: 3) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.04      0.09     0.00     0.26 1.00     1788
sd(weeks)                0.03      0.04     0.00     0.14 1.00     1550
cor(Intercept,weeks)     0.13      0.61    -0.95     0.98 1.00     3108
                     Tail_ESS
sd(Intercept)            1735
sd(weeks)                1331
cor(Intercept,weeks)     2857

~trial_no (Number of levels: 1821) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.13      0.00     0.12     0.14 1.00     2587
sd(weeks)                0.04      0.00     0.04     0.04 1.01      756
cor(Intercept,weeks)    -0.27      0.05    -0.35    -0.17 1.01      740
                     Tail_ESS
sd(Intercept)            4306
sd(weeks)                1659
cor(Intercept,weeks)     1947

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept     0.87      0.05     0.79     0.93 1.00     2669     1896
weeks         0.10      0.02     0.05     0.14 1.00     1761     1336

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.23      0.00     0.23     0.24 1.00     4476     3652

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

REMoxTB: Linear 30-day posterior

REMoxTB: Linear 42-day

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: log10(dtp_42) | cens(censored_42) ~ weeks + (1 + weeks | trial_no + treat) 
   Data: df_analysis_remox (Number of observations: 14734) 
  Draws: 4 chains, each with iter = 4000; warmup = 2000; thin = 1;
         total post-warmup draws = 8000

Group-Level Effects: 
~treat (Number of levels: 3) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.06      0.16     0.00     0.43 1.01      536
sd(weeks)                0.03      0.06     0.00     0.18 1.01      639
cor(Intercept,weeks)     0.14      0.61    -0.95     0.98 1.00      880
                     Tail_ESS
sd(Intercept)             361
sd(weeks)                 769
cor(Intercept,weeks)     1711

~trial_no (Number of levels: 1821) 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(Intercept)            0.13      0.01     0.13     0.14 1.01      404
sd(weeks)                0.04      0.00     0.04     0.05 1.03      214
cor(Intercept,weeks)    -0.23      0.05    -0.32    -0.13 1.04      100
                     Tail_ESS
sd(Intercept)            1667
sd(weeks)                 735
cor(Intercept,weeks)      175

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept     0.87      0.08     0.77     1.01 1.02      314      254
weeks         0.10      0.03     0.06     0.16 1.00      574      508

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.25      0.00     0.25     0.26 1.01      974     1751

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

REMoxTB: Linear 42-day posterior

REMoxTB: Nonlinear 25-day

REMoxTB: Nonlinear 25-day posterior

REMoxTB: Nonlinear 30-day

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: log10(dtp_30) | cens(censored_30) ~ alpha + beta1 * weeks + beta2 * gamma * log((exp((weeks - kappa)/gamma) + exp(-(weeks - kappa)/gamma))/(exp(kappa/gamma) + exp(-kappa/gamma))) 
         alpha ~ 1 + (1 | trial_no + treat)
         beta1 ~ 1 + (1 | trial_no + treat)
         beta2 ~ 1 + (1 | trial_no + treat)
         kappa ~ 1 + (1 | trial_no + treat)
         gamma ~ 1 + (1 | trial_no + treat)
   Data: df_analysis_remox (Number of observations: 14734) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~treat (Number of levels: 3) 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(alpha_Intercept)     0.02      0.01     0.00     0.06 1.47        8       22
sd(beta1_Intercept)     0.51      0.85     0.00     2.36 2.83        5       13
sd(beta2_Intercept)     0.48      0.50     0.00     1.84 2.75        5       16
sd(kappa_Intercept)     2.03      2.20     0.17     7.76 2.65        5       13
sd(gamma_Intercept)    21.73     18.17     0.03    63.99 3.12        5       11

~trial_no (Number of levels: 1821) 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(alpha_Intercept)     0.12      0.00     0.11     0.12 1.31       11       32
sd(beta1_Intercept)     0.03      0.00     0.02     0.03 2.92        5       30
sd(beta2_Intercept)     0.22      0.15     0.01     0.51 1.86        6       16
sd(kappa_Intercept)     0.69      0.44     0.07     1.43 3.93        4       11
sd(gamma_Intercept)     4.71      5.31     0.01    15.57 2.81        5       27

Population-Level Effects: 
                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
alpha_Intercept     0.83      0.02     0.79     0.85 2.38        5       31
beta1_Intercept     0.55      0.76     0.07     2.09 2.36        5       12
beta2_Intercept    -0.24      0.35    -1.05     0.42 2.00        5       11
kappa_Intercept     5.00      3.00     2.10    10.51 2.49        5       11
gamma_Intercept     1.09      0.56     0.17     1.91 2.06        5       48

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.23      0.00     0.22     0.23 2.54        5       15

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

REMoxTB: Nonlinear 30-day posterior

REMoxTB: Nonlinear 42-day

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: log10(dtp_42) | cens(censored_42) ~ alpha + beta1 * weeks + beta2 * gamma * log((exp((weeks - kappa)/gamma) + exp(-(weeks - kappa)/gamma))/(exp(kappa/gamma) + exp(-kappa/gamma))) 
         alpha ~ 1 + (1 | trial_no + treat)
         beta1 ~ 1 + (1 | trial_no + treat)
         beta2 ~ 1 + (1 | trial_no + treat)
         kappa ~ 1 + (1 | trial_no + treat)
         gamma ~ 1 + (1 | trial_no + treat)
   Data: df_analysis_remox (Number of observations: 14734) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~treat (Number of levels: 3) 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(alpha_Intercept)     0.03      0.11     0.00     0.28 1.33       10       48
sd(beta1_Intercept)     0.01      0.04     0.00     0.04 1.39        9       20
sd(beta2_Intercept)     0.27      0.48     0.00     1.81 2.60        5       27
sd(kappa_Intercept)     0.98      1.21     0.07     3.92 2.71        5       23
sd(gamma_Intercept)    14.39     10.41     0.04    38.32 2.61        5       12

~trial_no (Number of levels: 1821) 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(alpha_Intercept)     0.12      0.01     0.11     0.14 1.79        6       20
sd(beta1_Intercept)     0.03      0.00     0.03     0.04 2.23        5       20
sd(beta2_Intercept)     0.19      0.10     0.02     0.35 2.43        5       11
sd(kappa_Intercept)     0.60      0.46     0.17     1.45 2.88        5       24
sd(gamma_Intercept)     5.03      3.67     0.01    10.48 3.27        4       18

Population-Level Effects: 
                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
alpha_Intercept     0.83      0.04     0.71     0.86 1.81        6       26
beta1_Intercept     0.13      0.05     0.10     0.24 2.59        5       21
beta2_Intercept    -0.25      0.11    -0.43    -0.10 2.12        5       16
kappa_Intercept     3.18      0.52     2.08     3.80 2.04        6       17
gamma_Intercept     1.12      0.49     0.23     1.83 1.56        7       20

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.25      0.00     0.24     0.25 1.32       10       51

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

REMoxTB: Nonlinear 42-day posterior